Problem 32 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} 26 x-10 y &=-4 \\ -0.6 x+1.2 y &=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \((1, 3)\).

Step 1: Simplify the Equations

First, let's rewrite each equation to make it easier to work with. The first equation is already simplified: \( 26x - 10y = -4 \). For the second equation, multiply through by 10 to eliminate the decimal, giving us \( -6x + 12y = 30 \).

Step 2: Align for Elimination

We will use elimination to solve the system. First, let's rewrite the system for clarity: \( 26x - 10y = -4 \) and \( -6x + 12y = 30 \). To eliminate one variable, we'll align these equations and find a common multiple for elimination.

Step 3: Eliminate Variable x

Multiply the second equation by 13 and the first equation by 3 to facilitate elimination of \(x\): \[ \begin{align*} 3(26x - 10y) &= 3(-4) \quad \Rightarrow \quad 78x - 30y = -12 \ 13(-6x + 12y) &= 13(30) \quad \Rightarrow \quad -78x + 156y = 390 \end{align*} \]

Step 4: Add Equations to Eliminate x

Now add the two equations: \[ \begin{align*} 78x - 30y &= -12 \ -78x + 156y &= 390 \ \ ext{Add:} \ 0x + 126y &= 378 \end{align*} \] This simplifies to \(126y = 378\).

Step 5: Solve for y

Divide both sides by 126 to solve for \(y\): \[ y = \frac{378}{126} = 3 \] So, \(y = 3\).

Step 6: Substitute y Back to Find x

Substitute \(y = 3\) back into the first equation: \(26x - 10(3) = -4\), which simplifies to \(26x - 30 = -4\). Add 30 to both sides to get: \(26x = 26\).

Step 7: Solve for x

Divide both sides by 26 to solve for \(x\): \[ x = \frac{26}{26} = 1 \] So, \(x = 1\).

Step 8: Conclusion

The solution of the system is \(x = 1\) and \(y = 3\). Written as an ordered pair, the solution is \((1, 3)\).

Key Concepts

Elimination Method

In solving systems of equations, the elimination method is a powerful tool. It's great when you want to remove one variable by adding or subtracting the equations. Here is a simple breakdown of how it works:

• First, look at your system of equations and identify a variable to eliminate. This could be either the 'x' or 'y' variable.
• Then, manipulate the equations such that the coefficients of that variable are opposites. You can do this by multiplying one or both equations by suitable numbers.
• Once the coefficients are opposites, add or subtract the equations to eliminate that variable completely.

In our case, we eliminated 'x' by aligning the coefficients through strategic multiplication. After elimination, you're left with a simpler equation in one variable, making it straightforward to solve for that variable. In a nutshell, the elimination method streamlines the problem-solving process and gets you closer to finding the solution.

Solution of Linear Systems

A linear system consists of two or more linear equations. The solution describes where these lines intersect, meaning where they share the same set of values for their variables. Here's how you can find the solution:

• Use either the elimination method or substitution. As we've seen above, elimination is useful when you want to quickly remove one variable by aligning coefficients.
• After one variable is eliminated, solve the remaining equation for the second variable.
• Substitute the found variable value back into one of the original equations to find the other variable.

This step-by-step approach ensures you reach the intersection point of the two lines, which is your solution. If done correctly, you'll either find a single intersection point (one solution), identify parallel lines (no solution), or discover coincident lines (infinitely many solutions). In our example, we found a unique solution.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially when discussing coordinates and solutions in algebra. An ordered pair is just two numbers showing a specific point on a plane, often written as \(x, y\).

• The first number represents the horizontal position or the 'x' coordinate.
• The second number represents the vertical position, or the 'y' coordinate.

When solving a system of equations, finding an ordered pair is like finding the exact spot where the lines meet on a graph. Ordered pairs simplify the communication of a solution, succinctly conveying the values of 'x' and 'y'. In our problem, the solution was expressed as \(1, 3\), indicating that when \x = 1\, \y = 3\ satisfies both equations. This ordered pair is the key to understanding the solution physically on a coordinate grid.